Find the value of $\lim_{x \to 81} \frac{x^2 - 81^2}{x - 81}$.
Using L'Hopital's rule or factoring: $(x^2 - 81^2)/(x - 81)$ = x + 81. At x = 81, limit = 162.
What is the derivative of $f(x) = x^82$ at x = 1?
f'(x) = 82 $x^{81}$. At x = 1, f'(1) = 82.
Consider the function f(x) = (x - $83)^2$. Which of the following statements are TRUE?
f(83) = 0, f'(83) = 0, f''(83) = 2 > 0 (so x = 83 is a local MINIMUM, not maximum).
Find the value of $\lim_{x \to 84} \frac{x^2 - 84^2}{x - 84}$.
Using L'Hopital's rule or factoring: $(x^2 - 84^2)/(x - 84)$ = x + 84. At x = 84, limit = 168.
What is the derivative of $f(x) = x^85$ at x = 1?
f'(x) = 85 $x^{84}$. At x = 1, f'(1) = 85.
